sequence

A sequence is just an ordered list of numbers, going on and on: a first one, a second one, a third one, with no end. Think of the readings on a thermometer taken every minute, or the amount left in a battery as it slowly drains. The order matters and each spot in the line has its own number.

We write a sequence as a_1, a_2, a_3, ... where a_n is the n-th term. A sequence converges to a limit L if its terms settle down toward L as you go further out — formally, for every tolerance epsilon > 0 there is a point N beyond which |a_n - L| < epsilon. If the terms never settle on a single finite value, the sequence diverges. For example a_n = 1/n converges to 0, while a_n = n grows without bound and diverges.

It is easy to confuse a sequence with a series, but they are different things. A sequence is the list of terms themselves; a series is what you get when you add those terms up. The sequence 1/2, 1/4, 1/8, ... has terms marching toward 0, yet the SUM of those same terms is a separate question with its own answer.